In [Proposition 2.7, [1]}], Shimorin remarkably observed that \(T\)  and its Cauchy dual \(T^{\prime }\)  has the following dual properties:
\(\bigvee _{m \in \mathbb {N}} T^m \ker T^* = (\underset{m \ge 1}{\bigcap }T^{\prime m} \mathcal {H})^{\perp }; \bigvee _{m \in \mathbb {N}} T^{\prime m} \ker T^* = (\underset{m \ge 1}{\bigcap }T^m \mathcal {H})^{\perp }.\) 
